Question

1. please complete the formal proof

given: (angle 1congangle 2)
(overline{zn}congoverline{ei})
e midpoint of (overline{hi})
prove: henz is a parallelogram

Explanation:

Step1: Definition of mid - point

Since \(E\) is the mid - point of \(\overline{HI}\), we have \(HE = EI\).

Step2: Angle - Side - Angle (ASA) congruence

We know \(\angle1\cong\angle2\), \(\overline{ZN}\cong\overline{EI}\) (given), and \(HE = EI\) (from step 1). Also, \(\angle HEN=\angle IZN\) (alternate interior angles for lines \(ZN\) and \(HE\) with transversal \(EN\)).
By ASA congruence, \(\triangle HEN\cong\triangle IZN\).

Step3: Corresponding parts of congruent triangles

From \(\triangle HEN\cong\triangle IZN\), we get \(ZN = HE\) and \(EN=ZN\) (corresponding parts of congruent triangles).

Step4: Parallelogram property

A quadrilateral \(HENZ\) is a parallelogram if both pairs of opposite sides are congruent. Since \(ZN = HE\) and \(EN = ZH\) (from step 3), \(HENZ\) is a parallelogram.

Answer:

The quadrilateral \(HENZ\) is a parallelogram as both pairs of opposite sides (\(ZN\) and \(HE\), \(EN\) and \(ZH\)) are congruent.